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Phenomenological analysis of lepton and quark Yukawa couplings in SO(10) two Higgs model

Koichi MATSUDA

Graduate school of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan

arXiv:hep-ph/0401154v1 21 Jan 2004

Abstract

We investigate a model that the Yukawa coupling form is constructed by two kinds of matrix (M0 and M1 ). For example, in the SO(10) GUT model, M0 and M1 are Yukawa couplings generated by the 10 and 126 Higgs scalars. We study how this model can give the observed mass and mixings of quarks and leptons. Parameter ?tting is fully scanned by assuming all the input data to be normally distributed around the center value. PACS number(s): 12.15.Ff, 12.10.Kt, 14.60.Pq

Typeset using REVTEX 1

I. INTRODUCTION

The grand uni?cation theory (GUT) is very attractive as an uni?ed description of the fundamental forces in the nature. However, in order to reproduce the observed quark and charged-lepton masses and mixings, a lot of Yukawa couplings are usually brought into the model. We think that the nature is simple. So it is the very crucial problem to know the minimum number of Yukawa couplings which can give the observed fermion mass spectra and mixings. However, if the quark and lepton Yukawa couplings are composed by only one matrix Mu = cu M0 , Md = cd M0 , Me = ce M0 , (1.1)

the CKM Matrix must be diagonalized, these model is obviously ruled out for the description of realistic quark and lepton mass spectra. Therefore, at the uni?cation scale ? = ΛX , we

0 0 assume the Yukawa coupling of up quark, down quark and charged-lepton (Mu , Md and 0 Me ) are composed by two matrices,

0 0 0 Mf = cf 0 M0 + cf 1 M1 .

(f = u, d, e)

(1.2)

Here, cf 0 and cf 1 are real numbers which can be associated with the vacuum expectation values (VEV). For example, in the SO(10) GUT model with one 10 and one 126 Higgs scalars, the Yukawa couplings of quarks and charged leptons are expressed in the following forms [1] [2]:

0 0 0 0 0 0 0 0 0 Mu = c0 M0 + c1 M1 , Md = M0 + M1 , Me = M0 ? 3M1 .

(1.3)

0 0 where M0 and M1 are symmetric Yukawa couplings. In the previous paper [1], eliminating 0 0 M0 and M1 from Eq.(1.2), we obtain the relation

0 0 0 0 0 Me = cu Mu + cd Md ≡ cu (Mu + κMd ),

(1.4)

where 2

cd =

cu0 ce1 ? ce0 cu1 cu0 cd1 ? cd0 cu1

and

cu =

ce0 cd1 ? cd0 ce1 . cu0 cd1 ? cd0 cu1

(1.5)

These relations are realized at the GUT scale, but each value of the Yukawa couplings is given by the experiment at the weak scale ? = mZ . Therefore, we must investigate how the mass ratios and CKM matrix parameters change from ? = ΛX down to ? = mZ . [3] In this paper, we distinguish between the values at ? = ΛX and ? = mZ by using the superscript ”0” or not.

II. NUMERICAL STUDY

0 0 0 Because Mu , Md , and Me are symmetric at the uni?cation scale ? = ΛX in the model 0 0 with one 10 and one 126 Higgs scalars, they are diagonalized by unitary matrices Uu , Ud , 0 and Ue , respectively, as 0? 0? 0 0? 0 0 0? 0 0? 0 0? 0 Uu Mu Uu = Du , Ud Md Ud = Dd and Ue Me Ue = De , 0 0 0 where Du , Dd , and De are diagonal matrices which are given by 0 Du ≡

(2.1)

1 + tan β ?2 1 + tan β 2 0 diag(m0 , m0 , m0 ) , Dd ≡ diag(m0 , m0 , m0 ), u c t d s b v2 v2 2 1 + tan β 0 diag(m0 , m0 , m0 ) , De ≡ e ? τ v2

(2.2)

Here, v(= 174 GeV) is VEV of Higgs , and it is divided into up and down quark (neutrino and charged lepton) in the ratio tan β. Using the Cabibbo-Kobayashi-Maskawa (CKM)

0? 0 matrix Vq0 which is expressed as Vq0 = Uu Ud , the relation (1.4) is rewritten as follows: 0? 0 0 0? 0 0 0 (Ue Uu )? De (Ue Uu )? = cu Du + cd Vq0 Dd Vq0T = cu (Du + κVq0 Dd Vq0T ).

(2.3)

We take a basis on which the up-quark Yukawa coupling is diagonal in order to compare with the experiment values and obtain the independent two equations: A (κ) ≡ m0 m0 e ?

2

+ m0 m0 ? τ + m0 ?

2 2

2

+ (m0 m0 ) τ e (m0 )2 τ

2

2

2 [Tr {Hq (κ)}]2 {Tr (Hq (κ))}2 ? Tr (Hq (κ))2

→1

(m0 )2 e B (κ) ≡ (m0 )2 e

+

m0 m0 m0 e ? τ + m0 ?

2

+

(m0 )2 τ

[Tr {Hq (κ)}]3 →1 3 det {Hq (κ)} 3

(2.4)

Here Hq (κ) is the following hermite matrix which is de?ned by the Yukawa couplings of quark:

0 0 0 0 Hq (κ) ≡ (Du + κV 0 Dd V 0? )(Du + κV 0 Dd V 0? )? .

(2.5)

If we ?nd the κ which sets A(κ) and B(κ) to 1 simultaneously, the three Yukawa couplings

0 0 0 Mu , Md and Me can be uni?ed into two matrices. However we don’t know precisely how

to determine these data, especially quark masses. And above procedures depend on these ambiguities. So in this paper, we substitute the random numbers which becomes following normal distributions [4]: |mu (2GeV)| = 2.9 ± 0.6MeV, |ms (2GeV)| = 80 ? 155MeV, mb (mb ) = 4.0 ? 4.5GeV, |md (2GeV)| = 5.2 ± 0.9MeV, |mc (mc )| = 1.0 ? 1.4GeV, mdirect = 174.3 ± 5.1GeV, t (2.6) (2.7) (2.8)

mpole = 0.510998902 ± 0.000000021MeV, e mpole = 105.658357 ± 0.00005, mpole = 1776.99 ± 0.29MeV τ ?

(2.9) (2.10)

sin θ12 = 0.2229 ± 0.0022, sin θ13 = 0.0036 ± 0.0007,

sin θ23 = 0.0412 ± 0.0020, δ = (59 ± 13)?

(2.11) (2.12)

for each mass and CKM mixing parameter 10,000 times. And we estimate the evolution e?ect about the values in Eqs. (2.6) - (2.12) from ? = mZ to ? = ΛX by using of RGE. [3] In this work, we suppose MSSM for tanβ = 10. Without loss of generality, we can make the masses of third generation positive real number. Although the remaining masses are complex under the ordinary circumstances, we assume that all masses are real in order to simplify the problem. Therefore, there are 16 combinations of the signs of the masses as shown in table I. As shown in Fig.1, we scan the range A(κ) = 1 by changing Im(κ) from -100 to 100 at 2000 equal intervals. Moreover, we get the maximum and minimum of B(κ) on the line of A(κ) = 1 4

by changing Im(κ) at 5000 equal intervals. Because B(κ) is continuous, there is the κ which sets A(κ) and B(κ) to 1 simultaneously when Min(B(κ)) < 1 < Max(B(κ)) as explained in Fig.2. In this way, we draw the histograms in Figs 3,4,5 and 6 which show the distribution of input values conforming to the requirements Min(B(κ)) < 1 < Max(B(κ)). The each summation of the conforming case is tabulated in Table 2 after the 10,000 substitutions. Expressed in another way, Table 2 shows the number of dots in the white area in Fig.2. In Fig. 7, each circle in the complex plane shows the value of κ to meet the requirement A(κ) = B(κ) = 1, and the total number of circle in each ?gure corresponds to the number in Table 2, obviously. From these ?gures and tables, it is understandable that the sign of mu is not important. Perhaps the reason is that mu is very small, and it is almost negligible in comparison with other masses.

III. CONCLUSION AND DISCUSSION

In conclusion, we have discussed the probability that the following model will be realized without ?ne tuning. The random numbers which become normal distributions have been substituted for each physical value at ? = mZ . And we have taken the RGE e?ect between ? = mZ and ΛX into consideration. In this way, the search for κ which sets A(κ) and B(κ) to 1 simultaneously has been repeated 10,000 times. By this way, we have arrived at three conclusions: (1) The probability that the model will be realized without ?ne tuning is about 5% if we select the appropriate signs (14) or (15) of the masses. (2) This probability will increase if the signs of md , and ms are same. This gives the suggestion to the texture model. For example, a model with a texture (Md )11 = 0 on the nearly diagonal basis of the up-quark Yukawa coupling Mu is denied because these model leads to md /ms < 0. (3) From Fig.3-Fig.6, this probability will increase if we make ms somewhat larger or smaller than the present experiment value properly. In the present paper, we have demonstrate that the quark and charged lepton Yukawa coupling can be uni?ed into only two matrices. However, we have not referred to the neutrino 5

masses and lepton ?avor mixings. The neutrino Yukawa coupling is given by

0 MD =

ce0 ce1 0 0 cu0 M0 + cu1 M1 , cd0 cd1

(3.1) (3.2)

0 0 0?1 0 Mν = c?1 MD M1 MDT . R0

Concerning this problem, we have not been able to ?nd the positive solutions within 3σ which is written by the paper [5] for the present. However, since there are many possibilities for the neutrino mass generation mechanism, we are optimistic about this problem.

ACKNOWLEDGMENTS

The author is grateful to Y.Koide, H.Fusaoka, T.Fukuyama, T.Kikuchi and H.Nishiura for the useful comments. This work is supported by the JSPS Research Fellowships for Young Scientists, No.3700.

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REFERENCES

[1] K. Matsuda, Y. Koide, and T. Fukuyama, Phys. Rev. D 64, 053015 (2001); K. Matsuda, Y. Koide, T. Fukuyama and H. Nishiura, Phys. Rev. D 65, 033008 (2002) [Erratumibid. D 65, 079904 (2002)]; T. Fukuyama and N. Okada, JHEP 0211, 011 (2002); T. Fukuyama and T. Kikuchi, Mod. Phys. Lett. A 18, 719 (2003). [2] K.S. Babu and R.N. Mohapatra, Phys. Rev. Lett. 70, 2845 (1993); D-G. Lee and R.N. Mohapatra, Phys. Rev. D 51, 1353 (1995); H.S. Goh, R.N. Mohapatra, Siew-Phang Ng, Phys. Lett. B 570, 215 (2003); H.S. Goh, R.N. Mohapatra, Siew-Phang Ng, Phys. Rev. D 68, 115008 (2003). [3] H. Fusaoka and Y. Koide, Phys. Rev. D 57, 3986(1998). [4] Particle Data Group, D.E. Groom et al., Eur. Phys. J. C 15, 1 (2002); M. Jamin et.al, Eur.Phys.J. C 24, 273 (2002). [5] M. Maltoni, T. Schwetz, M.A. Tortola and J.W.F. Valle, Phys. Rev. D 68, 113010 (2003).

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TABLES

(mu , mc , mt ) (0) (1) (2) (3) (4) (5) (6) (7) (+ + +) (? + +) (+ ? +) (? ? +) (+ + +) (? + +) (+ ? +) (? ? +) (md , ms , mb ) (+ + +) (+ + +) (+ + +) (+ + +) (? + +) (? + +) (? + +) (? + +) (8) (9) (10) (11) (12) (13) (14) (15) (mu , mc , mt ) (+ + +) (? + +) (+ ? +) (? ? +) (+ + +) (? + +) (+ ? +) (? ? +) (md , ms , mb ) (+ ? +) (+ ? +) (+ ? +) (+ ? +) (? ? +) (? ? +) (? ? +) (? ? +)

TABLE I. The combinations of the signs of (mu , mc , mt ) and (md , ms , mb ). The signs of the charged lepton are negligible in Eq.(2.4).

sum (0) (1) (2) (3) 344 328 225 209 (4) (5) (6) (7)

sum 34 30 35 35 (8) (9) (10) (11)

sum 56 60 54 56 (12) (13) (14) (15)

sum 283 294 470 482

TABLE II. The total number of the cases conforming to the requirements Min(B(κ)) < 1 < Max(B(κ)) after the 10,000 substitutions.

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FIGURES

FIG. 1. The relations in Eq.(2.4) on the complex plane of κ. The solid line show A(κ) = 1 and

0 the dotted line B(κ) = 1. This is an example which is given as follows: Du = diag(?4.5116 ? 10?6 , 0 0 ?0.0011789, 0.5028), Dd = diag(?5.6008 ? 10?5 , ?0.0007776, 0.038776), De = diag(1.8697 ? 10?5 , 0 0 0 0.0039461, 0.067375), θ12 = 0.22695, θ23 = 0.035057, θ31 = 0.0023936, and δ0 = 1.3173.

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FIG. 2. The maximum and minimum of B(κ) on the line of A(κ) = 1 in the case of (15) in Table 1. There are 10000 dots in all area, and 482 dots in the white area as tabulated in Table 2.

FIG. 3. The histograms show the distribution of data values conforming to the requirements Min(B(κ)) < 1 < Max(B(κ)). Each number in parentheses show the signs of the mass in Table 1. We bins the data values into 20 equally spaced containers, and show the number of elements in each container as a bar graph. The vertical solid and dotted lines show the center value and range of error in Eqs.(2.6) - (2.12), respectively.

10

FIG. 4. The histograms show the distribution of data values as Fig.3.

11

FIG. 5. The histograms show the distribution of data values as Fig.3.

12

FIG. 6. The histograms show the distribution of data values as Fig.3.

13

FIG. 7. The distribution of κ in the complex plane. Each circle shows the value of κ to meet the requirement A(κ) = B(κ) = 1

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